Adsorption of light gases in covalent organic frameworks: comparison of classical density functional theory and grand canonical Monte Carlo simulations

Microporous and Mesoporous Materials 324 (2021) 111263

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Microporous and Mesoporous Materials
journal homepage: www.elsevier.com/locate/micromeso

Adsorption of light gases in covalent organic frameworks: comparison of
classical density functional theory and grand canonical Monte Carlo
simulations
Christopher Kessler 1 , Johannes Eller 1 , Joachim Gross, Niels Hansen ∗
Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany

ARTICLE

INFO

Keywords:
Covalent organic frameworks
Classical density functional theory
Grand canonical Monte Carlo
Adsorption

ABSTRACT
A classical density functional theory (cDFT) based on the PC-SAFT equation of state is proposed for the
calculation of adsorption equilibria of pure substances and their mixtures in covalent organic frameworks
(COFs). Adsorption isotherms of methane, ethane, n-butane and nitrogen in the COFs TpPa-1 and 2,3-DhaTph
are calculated and compared to results from grand canonical Monte Carlo (GCMC) simulations. Mixture
adsorption is investigated for the methane/ethane and methane/n-butane binary systems. Excellent agreement
between PC-SAFT DFT and GCMC is obtained for all adsorption isotherms up to pressures of 50 bar. The
cDFT formalism accurately predicts the selective accumulation of longer hydrocarbons for binary mixtures in

the considered COFs. This application shows substantial predictive power of PC-SAFT DFT solved in threedimensional geometries and the results suggest the method can in the future also be applied for efficient
optimization of force field parameters or of structural properties of the porous material based on an analytical
theory as opposed to a stochastic simulation.

1. Introduction
Covalent organic frameworks (COFs) are ordered nanoporous materials formed by covalent bonds between organic building blocks
composed of light elements, such as carbon, nitrogen, oxygen, and
hydrogen [1]. The materials are characterized by their large surface
area, high porosity and low molecular weights. Therefore, a broad
variety of applications has been envisioned, including gas storage and
separation, catalysis, sensing, drug delivery, and optoelectronic materials development [2–10]. Their bottom-up synthesis based on small
building blocks allows the design of porous materials possessing a large
variety of pore sizes and topologies. Similar to other porous materials
such as zeolites or metal organic frameworks (MOFs), the number of
hypothetical structures exceeds the ones synthesized so far by three
orders of magnitude [11]. Databases of curated structures [12–14]
and computational workflows that automatize molecular simulation
and analysis are being developed to screen materials for different
purposes [15–17].
In two-dimensional (2D) COFs the organic building blocks are
linked into 2D atomic layers that further stack via 𝜋-𝜋 interactions
to crystalline layered structures. The manner in which adjacent sheets
stack in this assembly process forming the crystalline material largely

influence their material properties including pore accessibility and, in
turn, adsorption capacity [18,19]. It is therefore unclear how representative idealized structural models can be compared to real COF
materials. This calls for an efficient computational approach that is
able to quantify the impact of structural variations on the adsorption
behavior. An established technique for this purpose are molecular
simulations, in particular Monte Carlo simulations in the grand canonical ensemble [20] (GCMC). Molecular simulation studies targeting

adsorption and/or diffusion in COFs have considered relatively small
adsorbate molecules such as helium, argon, hydrogen, methane, nitrogen or carbon dioxide [21–28], respectively, for which force fields can
be expected to reproduce the fluid properties with reasonable accuracy.
However, for CO2 -adsorption on all-silica zeolites it was shown that
computed Henry coefficients may differ by more than two orders of
magnitude across different CO2 force fields, in particular for zeolites
with more confined pore features, while different force fields yield
consistent predictions of Henry coefficients, when structures are less
confined [29]. In the case of COFs, containing significantly larger pore
sizes compared to zeolites, the impact of the stacking motifs of adjacent
layers in the structural model used in the simulations is expected to
influence the simulation outcome at least to the same extent as residual
discrepancies in the force fields used [21,26–28].

∗ Corresponding author.
E-mail address: (N. Hansen).
1
These authors contributed equally to this work.

/>Received 24 March 2021; Received in revised form 19 June 2021; Accepted 22 June 2021
Available online 1 July 2021
1387-1811/© 2021 The Author(s).
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Microporous and Mesoporous Materials 324 (2021) 111263

C. Kessler et al.

2. Computational details

To increase versatility of computational methods, a more efficient
alternative to molecular simulation could be classical density functional theory (cDFT) [30,31] which is also rooted in the framework
of statistical mechanics but relies on an inhomogeneous density profile compared to explicit atomistic molecular simulation. One of the
most common applications of cDFT is adsorption in homogeneous slit
pores with two opposing planar walls. The solid is thereby modeled
by an external field which commonly takes the form of a LennardJones 9-3 potential or a Steele potential. cDFT accurately predicts the
adsorption behavior when compared to GCMC simulations including
layering transitions [32] and capillary condensation [33]. The adsorption behavior of real unordered porous materials, however, is often not
well represented by the homogeneous slit pore model with one given
pore size. This is because of the ambiguous pore structures with often
unknown porosity, chemical composition and pore size distributions.
Therefore, cDFT models include heterogeneities [34], both in pore size

distribution and surface roughness/chemical heterogeneity, to compare
accurately to adsorption experiments. Ordered porous media, in turn,
are characterized by their regular pore structure and, thus, provide
a consistency test between cDFT and molecular simulations beyond
one-dimensional homogeneous slit pores.
A key ingredient of cDFT is the Helmholtz energy functional used
to describe the fluid–fluid interactions. Whereas the hard-sphere repulsion is often represented by a functional based on fundamental
measure theory [35–37], dispersive attractions are either treated by a
simple mean-field theory which ignores density correlations of the fluid
or non-local weighted density approximations in combination with
an underlying equation of state. Comparative computational studies
of the adsorption in ordered porous frameworks between cDFT and
GCMC simulations were performed by different groups; each utilizing different Helmholtz energy functionals. Guo and co-workers [38]
compared adsorption isotherms of noble gases in MOFs using a meanfield approach. Fu and Wu [39] assessed the performance of different
dispersive Helmholtz energy functionals from mean-field theory to
weighted density approximations with an empirical equation of state
for the adsorption of methane in MOFs.
The above mentioned Helmholtz energy functionals only consider
spherical molecules and are, thus, not suited for the description of
elongated chain-like molecules. Helmholtz energy functionals based on
the statistical associating fluid theory (SAFT), in turn, are capable of
accurately describing inhomogeneous systems of chain fluids and were
successfully applied to adsorption studies. Wertheim’s first order thermodynamic perturbation theory (TPT I) [40–43] is the foundation of
SAFT, originally introduced by Chapman, Jackson and co-workers [44–
47]. TPT I contains the formation of molecular chains as tangentially
bound spherical segments and is instrumental for the modeling of chain
fluids. We refer to the literature for detailed reviews about SAFT variants and their respective applications [48–52]. Mitchell and coworkers
applied a Helmholtz energy functional [53] based on SAFT for squarewell potentials with variable range (SAFT-VR) to the calculation of
pore size distributions of activated carbons from experimental nitrogen
adsorption isotherms [54]. The so-obtained pore size distribution is

then used for the prediction of n-alkane adsorption isotherms. Tripathi
and Chapman proposed an iSAFT Helmholtz energy functional for
hetero-segmented chains and investigated pure and mixed n-alkane
adsorption in graphite slit pores [55].
The present study uses a functional based on the perturbed-chain
statistical associating fluid theory (PC-SAFT) equation of state (EoS)
[56,57], which also utilizes a weighted density approximation [58].
This functional was already successfully applied to adsorption in onedimensional slit pores [33] and the calculation of surface tensions
and Tolman lengths [59]. We assess the PC-SAFT DFT model for
predicting adsorption in ordered three-dimensional COF frameworks.
We consider the adsorption of light gases in two typical COFs and
compare results from GCMC and cDFT. The results are discussed in light
of methodological differences of the two approaches.

2.1. Classical density functional theory
In this section, we summarize the fundamental equations of classical density functional theory and the application to adsorption in
COFs. Density functional theory is formulated in the grand canonical
{
}
ensemble at constant chemical potentials 𝝁 = 𝜇𝑖 , 𝑖 = 1, … , 𝜈 of all
species, volume 𝑉 , and temperature 𝑇 . The grand canonical potential
was shown to be a unique functional of the inhomogeneous density
{
}
profile 𝝆 (𝐫) = 𝜌𝑖 (𝐫), 𝑖 = 1, … , 𝜈 and can be expressed as
𝛺 [𝝆 (𝐫)] = 𝐹 [𝝆 (𝐫)] −

𝜈
∑
𝑖=1

∫

(
)
𝜌𝑖 𝜇𝑖 − 𝑉𝑖ext (𝐫) d𝐫

(1)

where 𝐹 [𝝆 (𝐫)] is the intrinsic Helmholtz energy functional capturing
the fluid–fluid interactions and 𝑉𝑖ext (𝐫) is the external potential due
to solid–fluid interactions acting on species 𝑖. For adsorption in microporous materials it is instructive to think of the system as being
connected to a large bulk reservoir with the same temperature and
chemical potentials 𝝁, so that a pressure of a communicating bulk fluid
𝑝(𝝁, 𝑇 ) can be calculated. The equilibrium density distribution 𝝆0 (𝐫)
minimizes the grand canonical functional
[
]
[
]
𝛺 𝝆 (𝐫) ≠ 𝝆0 (𝐫) > 𝛺 𝝆0 (𝐫) = 𝛺 (𝝁, 𝑉 , 𝑇 )
(2)
and its value is then equal to the grand canonical potential 𝛺 (𝝁, 𝑉 , 𝑇 ),
so that
𝛿𝛺 [𝝆] ||
=0
∀𝑖
(3)
𝛿𝜌𝑖 ||𝜌𝑖 (𝐫)=𝜌0 (𝐫)
𝑖

The equilibrium density profile is obtained by solving the Euler–
Lagrange equation
𝛿𝛺 [𝝆(𝐫)]
𝛿𝐹 [𝝆(𝐫)]
=
− 𝜇𝑖 + 𝑉𝑖ext (𝐫) = 0
𝛿𝜌𝑖
𝛿𝜌𝑖 (𝐫)

(4)

using a damped Picard iteration in combination with an Anderson
mixing scheme to accelerate the convergence rate [60].
For a compact notation, we henceforth omit the superscript 0 in
the equilibrium density profile; we use 𝝆 (𝐫) for the vector of density
profiles of all components in the system.
The intrinsic Helmholtz energy functional 𝐹 [𝝆(𝐫)] describes the
fluid–fluid interactions and is based on the PC-SAFT equation of state.
The coarse-grained molecular model of the PC-SAFT equation of state
represents molecules as chains of tangentially bound spherical segments. In this work, we only consider non-polar, non-associating
molecules, leading to the following Helmholtz energy contributions
𝐹 [𝝆(𝐫)] = 𝐹 ig [𝝆(𝐫)] + 𝐹 res [𝝆(𝐫)]
𝐹 res [𝝆(𝐫)] = 𝐹 hs [𝝆(𝐫)] + 𝐹 hc [𝝆(𝐫)] + 𝐹 disp [𝝆(𝐫)]

(5)

with repulsive hard-sphere interactions [36,37,61] (hs), hard-chain
formation [62,63] (hc), and van der Waals (dispersive) attraction of
chain fluids [57,58] (disp). The ideal gas contribution is exactly known
from statistical mechanics and reads

𝐹 ig [𝝆(𝐫)] = 𝑘B 𝑇

𝜈
∑
𝑖=1

∫

[ (
]
)
𝜌𝑖 (𝐫) ln 𝜌𝑖 𝛬3𝑖 − 1 d𝐫

(6)

with the Boltzmann constant 𝑘B and the de Broglie wavelength 𝛬𝑖 of
species 𝑖 containing intramolecular and kinetic degrees of freedom.
The White-Bear functional [36,37] is based on Rosenfeld’s fundamental measure theory [35] and is a commonly used Helmholtz energy
functional to model hard sphere repulsion. However, we find the functional inadequate for the description of fluids in the narrow cylindrical
pores encountered in the COF frameworks. Rosenfeld [61] presented
a modification to Helmholtz energy functionals based on fundamental measure theory for fluids in strong confinement that reduces the
effective dimensionality of the system. The resulting antisymmetrized
functional yields accurate results for hard spheres in narrow cylindrical
2


Microporous and Mesoporous Materials 324 (2021) 111263

C. Kessler et al.

pores, i.e. quasi one-dimensional systems, while retaining the full threedimensional properties and the bulk behavior of the original White Bear
functional. Additional details on the hard-sphere functional used in this
work are provided in the supplementary material.
The required pure component parameters for the utilized Helmholtz
energy contributions are the number of segments per molecule 𝑚𝑖 , the
segment size parameter 𝜎𝑖 and the dispersive energy parameter 𝜀𝑖 . We
here use an approach that does not capture the connectivity of the
different segments of a chain. Rather, the local density of segments 𝜌𝑖 (𝐫)
are considered as averages over all segments 𝛼𝑖 of the chain, as
𝜌𝑖 (𝐫) =

𝑚𝑖
1 ∑
𝜌 (𝐫)
𝑚𝑖 𝛼 𝛼𝑖

2.2. Grand canonical Monte Carlo simulation
All GCMC simulations were performed using the molecular simulation software RASPA [67]. Intramolecular fluid and intermolecular
fluid–fluid interactions were described with the TraPPE force field
[68,69]. The CHx groups in methane, ethane and 𝑛-butane were considered as single, chargeless interaction centers (united atoms) with
effective Lennard-Jones potentials. Parameters for unlike interaction
sites were determined using Lorentz–Berthelot combining rules. TraPPE
approximates the quadrupolar nature of nitrogen by placing negative
partial atomic charges at the position of the nitrogen atoms and a
neutralizing positive partial charge at the center of mass. The COF
framework was considered to be rigid such that only Lennard-Jones
parameters and partial atomic charges needed to be assigned to the
different atomic species. The Lennard-Jones parameters were taken
from the DREIDING force field [66]. Partial atomic charges of the
COF structures were calculated using the extended charge equilibration

(EQeq) [70] method implemented in RASPA. EQeq expands charge
equilibration (Qeq) [71] including measured ionization energies. The
method was tested for screening MOFs [72] and is computationally fast.
For the purpose of the present work, where partial charges play only a
minor role, this approach is sufficient. For other purposes an evaluation
of different variants of the algorithm [73] may be required or training
the algorithm for COFs [16,74]. Also test calculations using sophisticated methods such as REPEAT [75] or DDEC [76] which are based
on electronic structure calculations on the DFT level are recommended
to validate results from EQeq calculations. All force field parameters
applied in the present work are reported in the supplementary material. To be comparable to the classical DFT calculations described
in the previous section, the cut-off radius used for the Lennard-Jones
corrections was 14.816 Å , which is equal to four times the 𝜎CH4 parameter of the PC-SAFT EoS [57]. Although the density beyond the
cut-off radius is not uniform, we applied analytic corrections to the
long-range Lennard-Jones tail, in order to reduce the sensitivity of
the results with respect to the cut-off radius [77]. The real part of
the electrostatic interactions was evaluated up to a cut-off radius of
12.0 Å. Long-range electrostatic interactions were calculated by Ewald
summation [78,79] with a relative precision of 10−6 . To carry out
simulations at constant chemical potential, the PC-SAFT EoS was used
to pre-compute a fugacity coefficient at the given temperature and
pressure that was then passed to the MC code. The number of MC cycles
was 25 × 104 , both, for equilibration and for the production phase. One
cycle consists of max(20, 𝑁𝑡 ) MC moves (with 𝑁𝑡 as the sum of adsorbate
molecules in the system), i.e. translation, insertion or deletion and, in
case of molecules represented by more than one site, rotation moves.
In simulations of binary mixtures identity swap moves were carried out
additionally. All moves were performed with equal probability.

(7)

𝑖

leading to 𝜌𝑖 (𝐫) = 𝜌𝛼𝑖 (𝐫) for homosegmented chains.
The external potential represents the van der Waals interactions
exerted by the COF atoms onto a fluid (segment). The external potential
is calculated by considering the interactions of a PC-SAFT molecule
with all individual solid atoms of the framework, leading to
((
)12 (
)6 )
𝑀
∑
𝜎𝛼𝑖
𝜎𝛼𝑖
−
(8)
𝑉𝑖ext (𝐫) = 𝑚𝑖
4𝜀𝛼𝑖
|𝐫𝛼 − 𝐫 |
|𝐫𝛼 − 𝐫 |
|
|
|
|
𝛼=1
where 𝑀 is the number of solid atom interaction sites of the considered framework and 𝐫𝛼 is the position of the atom interaction site 𝛼
generated from the crystallographic information file (CIF) of the COF
framework. The interaction parameters 𝜀𝛼𝑖 and 𝜎𝛼𝑖 are calculated using
Lorentz–Berthelot combining rules [64,65] with
𝜎𝛼𝑖 = (𝜎𝛼 + 𝜎𝑖 )∕2

√
𝜀𝛼𝑖 = 𝜀𝛼 𝜀𝑖
where 𝜎𝛼 and 𝜀𝛼 are the Lennard-Jones interaction parameters of
atom interaction site 𝛼 taken from the DREIDING force field [66]
representing the COF structure.
In this work, the vector containing the number of adsorbed
{
}
molecules 𝐍ads = 𝑁𝑖ads , 𝑖 = 1, … , 𝜈 of a 𝜈 component mixture is
calculated with
𝐍ads =

∫

𝝆 (𝐫) d𝐫

(9)

using the vector of density profiles 𝝆 (𝐫) of all components in the system.
Similar to experiments, the fluid in the COF framework is in equilibrium with a bulk phase reservoir. The number of adsorbed molecules
𝐍ads in the COF framework can then be calculated from the bulk
conditions: for defined temperature 𝑇 , pressure 𝑝 and molar fractions
{
}
𝐱 = 𝑥𝑖 , 𝑖 = 1, … , 𝜈 of the bulk reservoir, we first calculate the
chemical potentials 𝝁(𝑝, 𝑇 , 𝐱) from the PC-SAFT equation of state, we
then use Eq. (4) for determining the equilibrium densities 𝝆(𝐫) and
subsequently obtain the adsorbed amount using Eq. (9). The density
profile is considered convergent if the L2-norm of the Euler–Lagrange
equation is less than the tolerance of 1.0 × 10−12 ,

2.3. COF structures

)‖
(
‖∑
res
‖ 𝜌𝑖 (𝐫)𝛬3 − exp 𝛽𝜇𝑖 − 𝛿𝛽𝐹 [𝝆(𝐫)] − 𝛽𝑉 ext (𝐫) ‖
‖
‖ 𝑖
𝑖
𝑖
𝛿𝜌
𝑖
‖2
‖
< 1.0 × 10−12
res =
√
𝑁𝑥 ⋅ 𝑁𝑦 ⋅ 𝑁𝑧 ⋅ 𝜈

The two COFs considered in the present work are the ketoenaminelinked COF TpPa-1 [80] and the imine-linked COF 2,3-DhaTph [81,82]
having pore sizes of approximately 1.8 and 2.0 nm, respectively, see
Fig. 1.
As the purpose of this study is the comparison between two computational approaches somewhat idealized structures were used. For the
hexagonal COF TpPa-1 we assumed a perfectly eclipsed arrangement,
with coordinates taken from Cambridge structural database [83] under deposition number 945096 [80]. A detailed investigation of the
effects of interlayer slipping on adsorption was for example reported
by Sharma et al. [27].
For the COF 2,3-DhaTph initial coordinates in a perfectly eclipsed

arrangement were taken from the CoRe COF database [12,13]. However, the layer–layer distance in that structure of 6.7 Å is much larger
than the experimentally reported value of 4.0 Å because the benzene
rings were rotated by 90◦ . Moreover, the lattice was not tetragonal

(10)
(
)−1
where 𝛽 = 𝑘B 𝑇
is the inverse temperature and 𝑁𝑥 ⋅𝑁𝑦 ⋅𝑁𝑧 is the total
number of grid cells. The equilibrium density profile is then used as the
initial density profile of the next adsorption/desorption step. For the
first calculation at the lowest pressure, we chose the ideal gas solution
of the Euler–Lagrange equation (4) as the initial density profile
(
)
𝜌0𝑖 (𝐫) = 𝜌bulk
exp −𝛽𝑉 ext (𝐫)
(11)
𝑖
where 𝜌bulk
is the corresponding bulk density of species 𝑖. Using this
𝑖
procedure, we follow the local minima of the Euler–Lagrange equation
and detect phase transitions, e.g. capillary condensation [33], between
the adsorption- and desorption branches.
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Microporous and Mesoporous Materials 324 (2021) 111263

C. Kessler et al.

Fig. 1. Structural representation of the covalent organic frameworks studied in the present work. (a) TpPa-1; (b) 2,3-DhaTph. Carbon, nitrogen, oxygen and hydrogen are represented
as cyan, blue, red and white spheres, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Vapor–liquid coexistence curves for (a) methane, ethane, n-butane, (b) nitrogen and (c) binary mixtures of methane/ethane and methane/𝑛-butane. For the binary mixtures
the equilibrium pressure is plotted over the methane mole fraction at 298 K. The symbols represent Gibbs-Ensemble Monte Carlo simulations, the lines PC-SAFT calculations.

as in the experimentally derived X-ray structure [81,82], but rather
orthorhombic. To avoid artificial adsorption of adsorbates between the
layers the benzene rings were rotated by approximately 30◦ resembling
the value in the experimental structure, which allowed to bring the
layers closer together to 4.0 Å in our computational model without
inducing steric clashes. In the GCMC simulations 9 layers were used for
TpPa-1 and 8 layers for 2,3-DhaTph, resulting in simulation box sizes of
3.06 and 3.2 nm in 𝑧-direction, respectively. For the rectangular box of
2,3-DhaTph, the other dimensions are 4.0028 and 3.259 nm and for the
hexagonal box of TpPa-1 the lengths are 4.5112 nm in each direction.
The number of framework atoms are 2592 and 1760 for TpPa-1 and
2,3-DhaTph, respectively. The CIF-files of the two structures used in
the present work are provided in the supplementary material.

Fig. 2 shows that vapor–liquid coexistence curves in the
temperature–density projection for nitrogen, methane, ethane and 𝑛butane obtained from Gibbs-ensemble [88,89] Monte Carlo simulations
do not exhibit significant deviations between the two methods. For the
vapor liquid equilibrium of the mixtures some deviations occur for the
methane/𝑛-butane system. These deviations in the vapor phase can be
attributed to rather significant deviations in vapor pressures observed
for the TraPPE force field [90]. For the mixture of methane/ethane
sampling of a stable two-phase region was difficult to establish with

Gibbs-ensemble Monte Carlo, because the vapor–liquid phase envelop
is rather small and the system is close to the mixtures’ critical point for
all relevant compositions. However, simulations at 199.93 K, reported
by Chakraborti and Adhikari [91] showed a good agreement with experiment for the saturated liquid phase but significant deviations in the
coexisting vapor phase, similar to the methane/𝑛-butane case studied
here. As shown below, these differences do not have a significant
impact on the adsorption equilibria in the considered pressure range.
Therefore, an attempt to use improved variants of the TraPPE force
field [92] was not pursued.

2.4. Ideal adsorbed solution theory
Adsorption isotherms of mixtures can be estimated from the pure
component isotherms using the ideal adsorbed solution theory (IAST)
[84]. In the present work the IAST equations were solved with the
pyIAST package [85]. To account for non-ideal behavior of the gas
phase at elevated pressure fugacities instead of pressures were employed in the IAST equations [86,87].

3.1. Pure component adsorption
The pure component adsorption isotherms are presented by plotting the average absolute amount adsorbed 𝑁 ads per mass of COFframework as function of the pressure in the external reservoir. The
statistical uncertainties in the GCMC results are in almost all cases
smaller than the symbol size. Fig. 3 shows adsorption isotherms of nitrogen, methane and ethane in COF 2,3-DhaTph at 298 K with varying
pressure up to 50 bar. For methane excellent agreement between cDFT
and GCMC is obtained over the entire pressure range. For ethane the
agreement between the two approaches is very good up to pressures
of 2 bar. At higher pressures cDFT slightly underestimates the amount
adsorbed. For nitrogen the agreement between GCMC and cDFT is
remarkable given that the force field contains three collinear partial
atomic charges to model the quadrupolar nature of the molecule while

3. Results and discussion

Before comparing adsorption isotherms predicted by cDFT and
GCMC we first investigate vapor–liquid equilibria to assess whether
the two approaches show deviations that may impact their comparability. Note that the segment size parameter 𝜎𝑖𝑖 and the dispersive
energy parameter 𝜀𝑖𝑖 used in PC-SAFT are different from the forcefield parameters used in the MC simulations, even for methane. Both
were independently adjusted to experimental data of pure compounds.
Results from both approaches are comparable, however, because pure
component parameters were adjusted to experimental data for phase
equilibria.
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Microporous and Mesoporous Materials 324 (2021) 111263

C. Kessler et al.

Fig. 3. Pure component adsorption isotherms of nitrogen, methane and ethane in COF
2,3-DhaTph at 298 K obtained from GCMC simulations and classical DFT calculations.

Fig. 4. Pure component adsorption isotherms of nitrogen, methane and 𝑛-butane in
COF TpPa-1 at 298 K obtained from GCMC simulations and classical DFT calculations.

the PC-SAFT model entering the cDFT calculations describes nitrogen
as non-polar, so that the van der Waals parameters effectively capture
the (mild) quadrupole moment of nitrogen. Of course, the influence
of the partial atomic charges on the adsorption behavior strongly
depends on the magnitude of the charges of the adsorbent-framework,
as reported for siliceous zeolites [93]. For the here considered COF 2,3DhaTph nitrogen adsorption isotherms with and without framework
partial charges show only minor differences (see Figure S7 in the
supplementary material).
Fig. 4 shows adsorption isotherms of nitrogen, methane and 𝑛butane in COF TpPa-1 at 298 K up to a pressure of 50 bar. As before,

excellent agreement between GCMC and cDFT is obtained for methane.
For 𝑛-butane some deviations occur in the pressure range between
103 and 104 Pa in which the shape of the cDFT isotherm is less
smooth compared to its GCMC counterpart, possibly due to the averaged segment-density according to Eq. (7). Based on these deviations,
it is interesting to investigate a cDFT functional, where connectivity of
segments is accounted for and the densities of individual segments are
calculated. The formalism was proposed by Jain and Chapman [94]
and has also been applied with the PC-SAFT DFT model in previous
work of our group [95]. For nitrogen the cDFT isotherm is slightly
lower than the GCMC one. Again, we tested the influence of the partial
charges on the framework atoms with respect to nitrogen adsorption
and found that the GCMC isotherm is in excellent agreement with the
cDFT isotherm when evaluated in a framework exempt from partial
charges (see supplementary material). The TpPa-1 framework is somewhat more polar than 2,3-DhaTph, if we use the sum of squared partial
charges 𝑞𝑖 as a measure for how polar a framework is. We thus regard
the sum of 𝑁𝑖 𝑞𝑖2 ∕𝑉 , where 𝑁𝑖 is the number of atoms of species 𝑖
in the simulation cell and 𝑉 its volume. For all of the four atomic
species (C, H, N, O) these values are higher for TpPa-1 compared to
2,3-DhaTph. Therefore, for frameworks with low to moderate charge
densities, adsorption of quadrupolar fluids may be approximated by
dispersion interactions alone. In summary, and in view of the fact that
no parameter is adjusted for relating the two modeling approaches, we
consider the overall agreement observed in Figs. 3 and 4 as good.

Fig. 5. Adsorption isotherms for the methane/ethane mixture in COF 2,3-DhaTph at
298 K and 𝑥bulk
= 0.6. The IAST results are based on fits to the pure component GCMC
CH4
isotherms.

mixture predicted by cDFT are in very good agreement with the GCMC
results. Only at higher pressure GCMC predicts a slightly larger amount
adsorbed of ethane, as can be expected from the results obtained from
the pure component isotherms discussed above. IAST is in very good
agreement with the GCMC results indicating that the adsorbed phase is
approximated well by an ideal solution. We note, however, that IAST
takes the results from GCMC simulations of pure substances as input.
The mixture adsorption of methane and 𝑛-butane was studied in
the COF TpPa-1 at methane mole fraction of 𝑥bulk
= {0.2, 0.4, 0.6, 0.8}.
CH4
Fig. 6 shows the case 𝑥bulk
=
0.8,
all
other
cases
are
presented in the
CH4
supplementary material. Due to the much stronger adsorption of 𝑛butane relative to methane we introduced a second 𝑦-axis in Fig. 6,
to better present the amount of adsorbed methane. For methane, cDFT
and GCMC are in very good agreement as can be expected from the
comparison of the pure methane isotherms discussed above. The 𝑛butane isotherms as predicted from cDFT underestimate the adsorbed
amount as compared to the GCMC results, similar to the behavior for
pure 𝑛-butane. For methane IAST predicts a slightly higher amount
adsorbed above pressures of 104 Pa.

3.2. Binary mixture adsorption
The binary mixture isotherms are presented by plotting the average

absolute amount adsorbed of each species as function of the total pressure in the external reservoir. The mixture adsorption of methane and
ethane was studied in the COF 2,3-DhaTph at methane mole fractions
of the reservoir mixture of 𝑥bulk
= {0.1, 0.4, 0.6, 0.8}. Fig. 5 shows the
CH4
case 𝑥bulk
=
0.6.
All
other
cases
are presented in the supplementary
CH4
material. The adsorption isotherms of methane and ethane in the

4. Conclusion
A classical DFT approach relying on a Helmholtz energy functional
based on the PC-SAFT equation of state was used to predict adsorption
equilibria of pure components and binary mixtures in covalent organic
frameworks (COFs). The results were compared to adsorption isotherms
5


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C. Kessler et al.

Appendix A. Supplementary data
Supplementary material related to this article can be found online
at />References

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Fig. 6. Adsorption isotherms for the methane/𝑛-butane mixture in COF TpPa-1 at
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from GCMC simulations (using the TraPPE force field for the fluids).
While the latter approach is rooted in a fully atomistic description
(within a united atom approximation for alkanes), cDFT employs a
coarser description of the fluid by means of an analytical equation of
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that cDFT is a powerful alternative to GCMC for studying adsorption

equilibria in porous materials. The advantage of this approach is its
analytical nature allowing the calculation of derivatives and, therefore,
optimization tasks with respect to force field parameters or structural
properties of the porous materials. The second aspect is in particular
relevant for COFs because the stacking motifs have a substantial impact
on the materials properties, including the adsorption behavior. The
current limitation of the cDFT approach is a lower coverage of the
chemical space compared to molecular simulations, in particular with
regard to polar molecules.
CRediT authorship contribution statement
Christopher Kessler: GCMC simulations, Analysis, Writing. Johannes Eller: DFT calculations, Analysis, Writing. Joachim Gross:
Conceptualization, Supervision, Writing. Niels Hansen: Conceptualization, Supervision, Writing, Project administration.
Acknowledgments
This work was funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) - Project-ID 358283783 - SFB
1333 and Project-ID 327154368 – SFB 1313. Monte Carlo simulations were performed on the computational resource BinAC at High
Performance and Cloud Computing Group at the Zentrum für Datenverarbeitung of the University of Tübingen, funded by the state of BadenWürttemberg through bwHPC and the German Research Foundation
(DFG) through grant no INST 37/935-1 FUGG.
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8